I nternat. J. Math. & Math. Sci.
Vol.
391
(1978)391-392
RESEARCH NOTES
ON THE POST-WIDDER TRANSFORM
RICHARD BELLMAN
Departments of Mathematics,
Electrical Engineering, and Medicine
University of Southern California
Los Angeles, California 90007 U.S.A.
(Received June 18, 1978)
i.
INTRODUCTION.
The Laplace transform
L(f)
I
f(t)e-Stdt
(l.i)
0
is one of the most powerful tools of analysis.
A complex inversion formula
exists which allows us to make use of the theory of a complex variable to
obtain asymptotic relations and expansions [i].
It is of theoretical interest
to obtain an inversion formula which depends only on real value.
Such a
formula is furnished by the post-Widder inversion formula [2].
This formula is not useful computationally because the inverse is an
unbounded operator.
A great care has to be exercised and different methods
have to be employed [3].
The usual method for establishing the post-Widder formula depends upon
Laplace’s asymptotic evaluation of an integral.
ferent route which permits many generalizations.
Here we shall follow a dif-
392
2.
R. BELLMAN
THE PARSEVAL-PLANCHEREL FORMULA.
Let F be the Fourier transform of f.
Then we have
tdt
f(t)e0
(2.1)
dx,
0
We see then that any linear operator which transforms
I
into e
-ixt
furnishes an inversion formula using the standard results for the inversion
of the fourier transform
mation.
3.
[4].
The post-Widder formula is one such transfor-
In Widder’s book, other transformations are given.
GENERALIZATIONS.
We see that there are several possible generalizations.
place, we can divide other transformations.
expansion has a Parseval relation.
In the first
In the second place, any orthogonal
If we take the cotinuous analogue, we
obtain an analogue of the Parseval-Plancheral forumla.
We thus have a method
for obtaining inversion formulas for general transforms.
REFERENCES
i.
Bellman, R. and K. L. Cooke. Differential-Difference Equations, Academic
Press, Inc., New York, 1963.
2.
Widder, D. V. The Laplace Transform, Princeton University Press, Princeton,
New Jersey, 1941.
3.
Bellman, R., R. Kalaba and J. Lockett. Numerical Inversion of the Lapl.a.c.e
Transform, American Elsevier Publishing Company, Inc., New York, 1966.
4.
Titchmarsh, E. C.
KEY WORDS AND PHRASES.
The Fourier Integral, Oxford, Clarendon Press, 1959.
Post-Widdr and
transform.
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
44A15, 44A10.